Mucociliary clearance is an important pulmonary defense mechanism that serves to remove inhaled substances from the lung. It depends upon the relationship between cilia, mucus and periciliary fluid. The mucociliary function is depressed by a variety of water soluble atmospheric pollutants such as SO₂ and NO₂ [1]. The presence of surfactant in the mucoserous lining of airways helps in increasing the mucus transport and has been investigated experimentally [2-5]. It was pointed out that surfactant caused relative increase in transport rate [2]. It was also showed that in presence of surfactant mucus transport is more [3,5]. Bronchial surfactant is essential for bronchoalveolar transport mechanisms including ciliary and non-ciliary mucus transport [4]. In [5], Rubin et al. showed that surfactant therapy appears to improve mucus clearability.

In the case of pulmonary diseases (cystic fibrosis, chronic bronchitis, etc.) excessive amount of mucus is formed in the respiratory tract, which is transported mainly by coughing or forced expiration. This transport also depends upon the depths of mucus and serous layers and the rheological properties of mucus [6]. Mucus transport in a cough machine has been studied by a group of investigators under external applied pressure gradient [6-13]. In [7,8], Scherer and Burtz conducted fluid mechanical experiments relevant to coughing, using air and liquid blown out of a straight tube by turbulent jet. They showed that the liquid transport decreases as the viscosity of liquid increases by assuming that the flow is quasi-steady and turbulent stress of air is equal to viscous stress in the liquid. In [9-11], King and co-investigators in their experiments have shown that the transport increases with the increase in the thickness of mucous gel air flow rate and with the decrease in its elastic modulus. It was observed that mucous gel transport in a simulated cough machine increases as the viscosity of serous layer simulates decreases [6,12,13].

It may be noted that no mathematical model is developed so far to explain the above experimental observations, particularly with surfactant as a sol phase layer. In view of this, in this paper, we present a quasi-steady state three layer laminar flow model (mucous gel as viscoelastic Voigt element, air and surfactant sol phase fluid as Newtonian fluids) for mucous gel transport in a cough machine simulating trachea by considering the surfactant sol phase as serous layer. Due to the presence of surfactant, the slip effects at the boundaries of the surfactant layer are taken into account in the model. It is assumed that the gel transport is caused by a time dependent pressure gradient due to mild forced expiration.

We consider the quasi-steady state simultaneous laminar flow of surfactant sol phase fluid, viscoelastic mucous gel and air in a rectangular channel, relevant to mucous gel transport in a cough machine simulating a model trachea. The flow assumed to be caused by a time dependent pressure gradient generated by air motion simulating mild forced expiration in trachea. The flow geometry is shown in Figure 1, where surfactant sol phase fluid mucous gel and air regions are indicated.

The equations governing the laminar flow of surfactant sol phase fluid, viscoelastic mucous gel and air under quasi-steady state condition can be written as follows:

Region I surfactant sol phase

Region II mucous gel

Region III air

where is the time, is the coordinate in the direction of the flow, is the co-ordinate perpendicular to fluid flow, is the pressure; are the velocity components of sol phase fluid, mucous gel and air in the flow direction; are their respective densities and viscosities, G is the elastic modulus of mucous gel and is the shear stress in the mucous gel layer; is the shear stress in the sol phase layer and is the shear stress in the air region. It is assumed that mucous gel behaves like a viscoelastic Voigt element whose constitutive equation is given by equation (3) [14].

Mild forced expiration is a short time phenomena and a time dependent pressure gradient is generated in trachea. Therefore, we assume that

and is given by

where is the time, T is the duration of mild forced expiration and is a constant (independent of time). The function is plotted in Figure 2 for various T.

Since initially there is no pressure gradient, one can assume that the velocities and stresses are zero, therefore, the initial conditions are

The boundary and matching conditions for the system (1) - (4) can be written as follows:

Boundary conditions:

Matching conditions:

In equation (7) the right hand side represents the slip velocity at the surface which is caused by the slipperiness of the surfactant sol phase. Similarly in equation (9), the second term on the right hand side represents slip velocity at the interface and thus the condition of the continuity of the velocities at the interface is still valid. and in equations (7) and (9) are called the slip coefficients [15]. The corresponding slip velocities increase as slip coefficients increase. In a particular case, when the conditions (7) and (9) reduce to usual no-slip conditions.

Solving the equations (1)-(4) along with the initial, boundary and matching conditions (6)-(10), the expressions for the velocity components can be found as the following.

here, denotes the differentiation of with respect to and the expressions for and are given by the following.

where

The Volumetric flow rates per unit thickness in each of the layer are

which after using equations (11)-(13) can be found as

(16)

The average flow rates in each layer can be defined as

which after using equations (16)-(18) can be written as

Where,

(22)

(23)

In a particular case, when mucus behaves as a Newtonian fluid i.e. the expressions for and reduce to

The effects of rheological properties of mucous gel and its thickness, viscosity and thickness of sol phase fluid, slipperiness caused by surfactant sol phase and air flow rate on mucous gel flow rate are shown by plotting the expressions for given by equation. (20) in Figures 3-6( after eliminating with the help of equation (21)). The values of various parameters are taken as in the following [6,10,16-20].

Diameter of model trachea.

Thickness of mucous gel.

Thickness of sol phase.

Viscosity of air.

Viscosity of mucous gel.

Viscosity of sol phase.

Elastic modulus of mucous gel (G):

.

In our calculation, we assume

and.

Figure 3 is a plot of mucous gel flow rate versus airflow rate for different From this figure, it is observed that the effect of elastic modulus depends upon the magnitude of the viscosity of mucous gel. For less viscous mucous gel, Figure 3(a) shows that the gel flow rate decreases as the elastic modulus increases. This implies that the flow rate decreases when mucous gel becomes more elastic, suggesting that the efficient transfer